I don't think that your batteries will care if you have or not a Cap in there, if you 've got one, it won't hurt.
But, you'll have to limit the charge to 1.44V per cell (5.76 V total) to avoid overcharging your cells and cooking them.
You could put the above mentioned current source in series with a voltage regulator (another 317 using 2 resistors) that has been set to provide 5.76V to power the batteries.
The voltage reg will never allow more than the required voltage to the batteries and the current source will maintain the charging at a limit for the batteries.
Transf > Bridge> Cap> Current Source > Voltage Reg > Batteries.
Hope that helps